On the equivalence of some eternal additive coalescents
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1020-1037.

Nous étudions dans ce papier les coalescents additifs. En utilisant leur représentation en tant que processus de fragmentation, nous prouvons que certains coalescents additifs éternels ont une loi absolument continue par rapport à la loi du coalescent additif standard sur n'importe quel intervalle de temps borné inférieurement.

In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

DOI : 10.1214/07-AIHP154
Classification : 60J25, 60G09
Mots-clés : additive coalescent, fragmentation process 
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Basdevant, Anne-Laure. On the equivalence of some eternal additive coalescents. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1020-1037. doi : 10.1214/07-AIHP154. http://archive.numdam.org/articles/10.1214/07-AIHP154/

[1] D. Aldous and J. Pitman. The standard additive coalescent. Ann. Probab. 26 (1998) 1703-1726. | MR | Zbl

[2] D. Aldous and J. Pitman. Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent. Probab. Theory Related Fields 118 (2000) 455-482. | MR | Zbl

[3] J. Berestycki. Ranked fragmentations. ESAIM Probab. Statist. 6 (2002) 157-175 (electronic). Available at http://www.edpsciences.org/10.1051/ps:2002009. | Numdam | MR | Zbl

[4] J. Bertoin. Lévy Processes. Cambridge Univ. Press, 1996. | MR | Zbl

[5] J. Bertoin. Eternal additive coalescents and certain bridges with exchangeable increments. Ann. Probab. 29 (2001) 344-360. | MR | Zbl

[6] J. Bertoin. Self-similar fragmentations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 319-340. | Numdam | MR | Zbl

[7] J. Bertoin. On small masses in self-similar fragmentations. Stochastic Process. Appl. 109 (2004) 13-22. | MR | Zbl

[8] J. Bertoin. Random Fragmentation and Coagulation Processes. Cambridge Stud. Adv. Math. 102. Cambridge University Press (2006). | MR | Zbl

[9] J. Bertoin and A. Rouault. Note sur les fragmentations. Private communication.

[10] J. D. Biggins. Martingale convergence in the branching random walk. J. Appl. Probab. 14 (1977) 25-37. | MR | Zbl

[11] B. Chauvin and A. Rouault. KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Related Fields 80 (1988) 299-314. | MR | Zbl

[12] S. N. Evans and J. Pitman. Construction of Markovian coalescents. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 339-383. | Numdam | MR | Zbl

[13] J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes. Springer, Berlin, 1987. | MR | Zbl

[14] A. E. Kyprianou. Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris' probabilistic analysis. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 53-72. | Numdam | MR | Zbl

[15] G. Miermont. Ordered additive coalescent and fragmentations associated to Levy processes with no positive jumps. Electron. J. Probab. 6 (2001) no. 14, 33 pp. (electronic). Available at http://www.math.washington.edu/~ejpecp/EjpVol6/paper14.abs.html. | MR | Zbl

[16] J. Neveu. Multiplicative martingales for spatial branching processes. In Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987) 223-242. Progr. Probab. Statist. 15. Birkhäuser, Boston, MA, 1988. | MR | Zbl

[17] M. Perman, J. Pitman and M. Yor. Size-biased sampling of Poisson point processes and excursions. Probab. Theory Related Fields 92 (1992) 21-39. | MR | Zbl

[18] K.-I. Sato. Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, 1999. | MR | Zbl

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